This semester will be about uses of easy combinatorial models to solve longstanding open questions in representation theory and topology. (It continues last semester's theme, but last semester is not a prerequisite.) The combinatorial models include honeycombs (certain planar graphs made up of hexagons) and the closely related hives, as well as puzzles (pictures in the plane made of equilateral triangles and rhombi). These models can be used to study the representation theory of GL(n,C) and the topology of Grassmannians (the space of k-planes in Cn). One application of these methods is the solution of the Hermitian eigenvalue problem: If you know the eigenvalues of two Hermitian matrices A and B, what can you say about the possible eigenvalues of A+B?

Text: Computational Topology by Herbert Edelsbrunner and John L. Harer

In many applications data (of numerical information) can be represented by a point set in an n-dimensional space.
Computational topology focuses on the computational aspects of topology, and applies topological tools to analysis of those data sets. Topological tools of simplicial homology, shape and discrete
morse theory will be presented.
Algorithms and software for the analysis of the topology, connectivity
and shape of point sets will be explored. Projects will be either a
software exploration implementation on a point data set of interest to
you or a presentation of recent papers in the area.

Prerequisite: Consent of department. This course is required for all
incoming graduate students. It is a lab course meeting for an
average of two hours per week throughout the semester for
one-semester-hour's credit.

This course provides practical training in the teaching of mathematics at
the pre-calculus level, how to write mathematics for publication, and treats
other issues relating to mathematical exposition.
Communicating Mathematics I and II are designed to provide training
in all aspects of communicating mathematics. Their overall goal is to teach
the students how to successfully teach, write, and talk about mathematics to
a wide variety of audiences. In particular, the students will receive
training in teaching both pre-calculus and calculus courses. They will
also receive training in issues that relate to the presentation of research
results by a professional mathematician. Classes tend to be structured to
maximize discussion of the relevant issues. In particular, each student
presentation is analyzed and evaluated by the class.

This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. It will cover the basic notions of group, ring, module, and field theory. Topics will include symmetric groups, the Sylow theorems, group actions, solvable groups, Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, modules over PIDs, vector spaces, field extensions, and finite fields.

The systematic use of chain complexes and long exact sequences originated in the setting of algebraic topology, but it has now found applications throughout many areas of mathematics. In this course, we will develop these tools in a modern, algebraic way (with a focus on the language of derived categories), and we will discuss various applications in algebra and topology, depending on the interests of the class. Possible topics for applications include group cohomology, Lie algebra cohomology, or sheaf cohomology.

We will cover some special topics with a view to applications.
Possible topics are: (i) stochastic differential equations;
(ii) the study of the first time when a stochastic process hits a given boundary; (iii) the Feynman-Kac formula, solving a partial differential equation in terms of stochastic integrals; (iv) a sampler from random matrices; (v) large scale behavior of correlated random variables. Applications in finance and in the study of physical phenomena will be discussed.

Prerequisites: Any one of Math 3355, 3903, 4031, or 4038, or their equivalent.

Text:

In this course we provide an introduction to theory behind the design of metal-dielectric crystals for the control of light. Here we develop basic ideas and intuition as well as introduce the mathematics and physics of spectral theory necessary for the rational design of metamaterial crystals. The course provides a self contained introduction as well as a guide to the current research literature useful for understanding the mathematics and physics of wave propagation inside complex heterogeneous media. The course begins with an introduction to Bloch waves in crystals and provides an introduction to local plasmon resonance phenomena inside crystals made from nobel metals. We then show how to apply these techniques to construct media with exotic properties. We provide the mathematical underpinnings for characterizing the interaction between surface plasmon spectra and Mie resonances and its effect on wave propagation. This understanding is necessary for designing structured media supporting backward waves and behavior associated with an effective negative index of refraction. We conclude by introducing multiscale techniques necessary for computing the effective dielectric properties of metamaterials and higher order corrections.

Iterative methods are fundamental to large scale scientific computations, where the demands on memory and flops render most direct method impractical. The emphasis of this course is on the motivation, derivation and analysis of some of the most important iterative algorithms. The following topics will be covered.

Prerequisites: Consent of department. A course in graph theory is strongly recommended.

Text: None

We study the various ways in which combinatorics and computation interact. We will study efficient algorithms for combinatorial problems such as matching in graphs, as well as heuristics for equivalence-free generation of various combinatorial objects, and the use of advanced computational tools to show (non)existence of combinatorial objects, including integer programming and positive semidefinite optimization, and algebraic tools such as Groebner basis computation. In addition to traditional homework problems, the course will include a few computational projects. Some of the lectures will be devoted to an introduction to SageMath, but other computer algebra software can be used by the students if they desire.

This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group, the homology groups) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., isomorphic groups). Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent.

The focus of this course will be on cohomology theory, dual to the homology theory developed previously. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications.

In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, we may pursue some of these connections, such as cohomology of groups or the De Rham theorem, or other elements of algebraic topology, such as homotopy theory.

How is knot theory related to theoretical physics? This course is a gentle introduction to the mathematical tools needed to understand the relationship between knot theory and Chern-Simons theory. First we will use differential forms to formulate Maxwell’s equations of electromagnetism. Then we introduce connections on vector bundles whose underlying structure group is from a list of certain Lie groups, and we use the connections and curvature to generalize Maxwell theory to the Yang-Mills equations. This in turn motivates Chern-Simons theory and its relationship to knot theory.

This semester will be about uses of easy combinatorial models to solve longstanding open questions in representation theory and topology. (It continues the theme of the last two semesters, but those semesters are not prerequisites for this course.) The combinatorial models include honeycombs (certain planar graphs made up of hexagons) and the closely related hives, as well as puzzles (pictures in the plane made of equilateral triangles and rhombi). These models can be used to study the representation theory of GL(n,C) and the topology of Grassmannians (the space of k-planes in Cn). One application of these methods is the solution of the Hermitian eigenvalue problem: If you know the eigenvalues of two Hermitian matrices A and B, what can you say about the possible eigenvalues of A+B?

Prerequisite: Consent of department. This course is required for all
incoming graduate students. It is a lab course meeting for an
average of two hours per week throughout the semester for
one-semester-hour's credit.

This is the second semester of the first year graduate algebra sequence. Topics will include field theory, Galois theory, basics of commutative algebra and algebras over a field, Wedderburn’s theorem, Maschke’s theorem, tensor products and Hom for modules, possibly some introduction to homological algebra or linear representations of finite groups if time permitted.

In the same way that Galois theory has its origins in studying the
symmetries of the solutions of a polynomial, differential Galois
theory grew out of the analogous problem for linear differential
equations. More precisely, differential Galois theory studies
extensions of differential fields, i.e., fields equipped with a
derivation (a map obeying the Leibnitz rule). As the name suggests,
there is a remarkable formal parallelism between the two theories.
In particular, there is a fundamental theorem of differential Galois
theory. Indeed, for appropriate differential field extensions E/F, the
differential automorphisms fixing F forms a group called the
differential Galois group G, and there is a one-to-one inclusion
reversing correspondence between closed subgroups of G and
intermediate differential field extensions. However, whereas
classical Galois groups are finite (or at worst profinite),
differential Galois groups are linear algebraic groups.

This class will provide an introduction to differential Galois theory.
We will discuss Picard-Vessiot extensions (the differential analogue
of Galois extensions) and give several different constructions of the
differential Galois group. We will prove the fundamental theorem and
consider when differential equations are solvable by quadratures--the
differential analogue of solvability by radicals. We will further
discuss the monodromy group of linear differential equations and its
relationship to the differential Galois group.

This course will cover the qualitative theory of ordinary differential
equations. Topics will include existence and uniqueness theory,
dependence on initial conditions, linear systems, monodromy,
stability, and Hamiltonian systems. Time permitting, we will give an
introduction to asymptotic analysis and the Stokes phenomenon.

A standard first course in functional analysis. Topics include Banach spaces, Hilbert spaces, Banach algebras, topological vector spaces, spectral theory of operators and the study of the topology of the spaces of distributions.

The course will start with an introduction to Brownian motion, and martingales. This would lead to the development of stochastic integrals and stochastic differential equations. We will study a deep and fundamental connection between stochastic differential equations and a class of partial differential equations.

This course presents basic methods to obtain a priori estimates for solutions of second order elliptic partial differential equations in both divergence and non-divergence forms . Topics covered include weak and viscosity solutions, Hopf and Alexandroff maximum principles, Harnack inequalities, De Giorgi-Nash-Moser regularity theory, continuity and differentiability of solutions. The course can be viewed as a continuation of MATH 7386 but no prior knowledge of PDEs is necessary.

Grading in the course will be based on a semester long group project.
Professor will provide a "real" problem and guide the students on how to model it, how to formulate it mathematically and analyze it, and how to build and analyze a numerical method for it and simulate it. The necessary software framework will be provided. The final product will be a write-up describing the project, as well as plots/results with discussion, and the code needed to run it.

Prerequisites:
Students should know basic elliptic PDE theory (e.g. Poisson equation, convection/diffusion equation, weak formulations, etc.) as well as some numerical methods for solving PDEs, e.g. finite difference or finite element methods, etc.; however, some discussion on numerical methods will be given in the class. Knowledge of continuum mechanics is a plus, but not required (necessary concepts will be reviewed).

The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, and graph minors. For more information see Math 7410.

This is an introduction to the theory of graph minors. We will discuss many problems of the following two types: determine all minor-minimal graphs that have a prescribed property; determine the structure of graphs that do not contain a specific graph as a minor. We will focus on connectivity and planarity.

Prerequisites: Math 7410 and 7490 (Matroid Theory) or permission of the department.

Text: Matroid Applications edited by Neil White (Chapter 6: The Tutte Polynomial and its Applications by Thomas Brylawski and James Oxley)

The theory of numerical invariants for matroids is one of many aspects of matroid theory having its origins within graph theory. Most of the fundamental ideas in matroid invariant theory were developed from graphs by Veblen, Birkhoff, Whitney, and Tutte when considering colorings and flows in graphs. This course will introduce the Tutte polynomial for matroids and will consider its applications in graph theory, coding theory, percolation theory, electrical network theory, and statistical mechanics.

Text: Algebraic Topology by A. Hatcher, Cambridge Univ. Press
(This is available for free download in pdf format or may be purchased as a book)

We will discuss the homology groups of topological spaces. To a topological space, one associates a sequence of abelian groups called their homology groups.
To a continuous map, one associates a sequence of group homomorphisms in a functorial way. One application of homology is the Brouwer fixed point theorem
which asserts any continuous map from an n-dimensional disk to itself has a fixed point. One also has a higher dimensional version of the Jordan curve theorem.
We will learn to calculate homology groups in a variety of ways. If time permits, we will begin to discuss cohomology as well.

The mapping class group of a surface is the group of its orientation preserving homeomorphisms (up to a certain equivalence relation). The study of mapping class groups is a classical subject which has exploded in the last decade.

In this course we will study the algebraic structure of mapping class groups and the detailed description of its individual elements. This includes the Nielsen – Thurston classification theorem, which gives a particularly nice representative for each element of the group. Furthermore, we will introduce spaces on which mapping class groups act, such as the complex of curves and Teichm&uumlller space. An important theme throughout the course will be the relationships between the geometry of these spaces, the algebra of the mapping class group, and the topology of the surface.

Characteristic classes are incredibly powerful and useful cohomological invariants of vector bundles. In this course, we will discuss vector bundles, fiber bundles and characteristic classes with a focus on bundles over smooth manifolds. Beginning with some basic examples and constructions, we will eventually move on to the problem of classifying vector bundles. Along the way, we will construct Stiefel-Whitney, Euler, Chern and Pontryagin classes.

One of the many interesting applications of topology is the Brouwer Fixed Point Theorem.
We will study proofs of the Brouwer Fixed Point Theorem, and show that it is equivalent to Sperner’s lemma and to the theorem that the Game of Hex does not end in a tie.
This will lead to applications such as questions in Economics on how to divide the rent of shared apartments or the existence of a Nash equilibrium.